= Solution
Use the <positive Laplace-Beltrami operator>, $\Delta=-\operatorname{div}\nabla$. A <flat torus> is the quotient $\mathbb R^d/\Lambda$, where $\Lambda$ is a full-rank <Euclidean lattice> and the metric descends from the <Euclidean metric>. Its <dual lattice> is
$$
\Lambda^*=\{\xi\in\mathbb R^d:\langle\xi,\ell\rangle\in\mathbb Z\text{ for every }\ell\in\Lambda\}.
$$
If $V$ is the <covolume> of $\Lambda$, the functions $\phi_\xi(x)=V^{-1/2}e^{2\pi i\langle\xi,x\rangle}$ descend to the torus. Direct differentiation gives the <spectrum of a flat torus>:
$$
\boxed{\Delta\phi_\xi=4\pi^2|\xi|^2\phi_\xi,\qquad
\operatorname{Spec}\Delta=\{4\pi^2|\xi|^2:\xi\in\Lambda^*\}.}
$$
The multiset includes one entry for each dual vector. In particular zero has multiplicity one. Writing $\Lambda=B\mathbb Z^d$ identifies the dual vectors with $B^{-T}m$, $m\in\mathbb Z^d$. Standard <Fourier series> on the unit cube then prove that these functions are an <orthonormal basis> of $L^2$ on the torus. If $\Delta f=\lambda f$, its <Fourier coefficients> satisfy $(4\pi^2|\xi|^2-\lambda)\widehat f(\xi)=0$. Thus every <eigenfunction> is a linear combination of the indicated shell of frequencies, with no additional <eigenvalues>. Compactness gives the discrete self-adjoint realization, whose <eigenfunctions> are smooth. For real functions the two characters at $\xi,-\xi$ become cosine and sine, giving the same real multiplicities. The opposite Laplacian sign would negate all displayed <eigenvalues> without changing the rigidity conclusion.
To prove two-dimensional <spectral rigidity>, let $L=\Lambda^*$ and recover its <Gram matrix> from its vector-length multiset. Let $a$ be the shortest nonzero length and choose $v\in L$ with $|v|=a$. This vector is primitive: a proper integer multiple would have a shorter lattice vector. Hence $L\cap\mathbb Rv=\mathbb Zv$. From the spectral multiset subtract exactly two occurrences of each length $ka$, $k=1,2,\ldots$. The remaining multiset consists precisely of the vectors not on this line, including correct multiplicities at coincident lengths. Its least length $b$ is the length of a shortest vector $w$ independent of $v$.
The pair $(v,w)$ is a <basis> of $L$. Otherwise take a lattice point in a nonzero coset of $\mathbb Zv+\mathbb Zw$ and reduce its two coefficients into $[-1/2,1/2]$. The resulting nonzero lattice vector $z=\alpha v+\beta w$ is not on $\mathbb Rv$, by primitivity of $v$. But
$$
|z|\leq\frac{|v|+|w|}{2}\leq b,
$$
with strict inequality: when both coefficients are nonzero the triangle inequality is strict for independent vectors, and when one vanishes its length is at most $b/2$. This contradicts the definition of $b$. Subtract an integer multiple of $v$ from $w$ and, if necessary, change its sign to arrange $0\leq c=\langle v,w\rangle\leq a^2/2$; minimality of $b$ makes this reduction possible without decreasing its length.
The <covolume> $A$ of $L$ is also spectral. The count of lattice vectors with length at most $R$ obeys
$$
\#(L\cap B_R)=\frac{\pi R^2}{A}+O(R).
$$
For example, translate a bounded fundamental parallelogram of diameter bound $D$: the union of tiles with centres in $B_R$ contains $B_{R-D}$ and lies inside $B_{R+D}$. Comparing their areas proves the leading coefficient. The <spectrum> supplies this count. Since $(v,w)$ is a lattice <basis>, its parallelogram area satisfies $A^2=a^2b^2-c^2$. Therefore
$$
\boxed{\operatorname{Gram}(v,w)=
\begin{pmatrix}a^2&\sqrt{a^2b^2-A^2}\\\sqrt{a^2b^2-A^2}&b^2\end{pmatrix}.}
$$
All three quantities are determined by the <spectrum>. Equal <Gram matrices> give an orthogonal map between the <dual lattices>; taking duals gives an orthogonal map between the original lattices, descending to a <Riemannian isometry> of the tori. \b[Isospectral flat two-tori are isometric.] This is the <two-dimensional lattice reconstruction from vector lengths>; it uses lengths with multiplicities, not merely the set of distinct lengths.
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